Automorphic forms constructed from Whittaker vectors
Let G be a semisimple Lie group of split rank 1 and \(\Gamma\) a discrete subgroup of finite covolume. Forming a generalized Poincaré series of weak Whittaker functions attached to the nonunitary principal series a meromorphic family M(s) of functions on \(\Gamma\) \(\setminus G\) is obtained that satisfy all axioms of automorphic forms except the condition of moderate growth. It is shown that the principal part at a pole \(s_ 0\), Re \(s_ 0\geq 0\) is square integrable and that for a fixed K-type all automorphic forms up to a finite dimensional subspace are obtained. For the trivial K-type all automorphic forms are derived this way. The Fourier coefficients of the M-series are computed and a functional equation is derived. Trying to generalize this to higher rank leads to the observation that in certain cases (proved here for \(G=SO(n,1)^ d)\) the moderate growth condition for automorphic forms is redundant. This may be called a Koecher principle. The conjecture is stated that a Koecher principle holds for all G of rank\(\geq 2\) and \(\Gamma\) irreducible.
scientific article; zbMATH DE number 4132268
scientific article